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We offer analytical solutions to classical percolation problems on hypergraphs with arbitrary vertex degree and hyperedge cardinality distributions. We introduce a generalization of the 2-core for hypergraph and we show that it can emerge in either a continuous or a hybrid percolation transition. We also define two different hypergraph cores related to the hyperedge cover and vertex cover problems on hypergraphs. We validate our analytical results with extensive numerical simulations.